Local Orientation Transitions to a Lying Helix State in Negative Dielectric Anisotropy Cholesteric Liquid Crystal

Abstract

Orientation transitions in a cholesteric liquid crystal (CLC) layer with negative dielectric anisotropy, under the influence of a non-uniform spatially periodic electric field created using a planar system of interdigitated electrodes, were studied experimentally and numerically. In the interelectrode space, transitions are observed from a planar Grandjean texture, with the helix axis perpendicular to the layer plane, to states with a lying helix, when the helix axis is parallel to the layer plane and perpendicular to the electrode stripes. It was found that the relaxation time of the induced state in the Grandjean zones, corresponding to two or more half-turns of the helix, significantly exceeded the relaxation time for the first Grandjean zone with one half-turn. An analysis of experimentally observed and numerically simulated textures shows that slow relaxation to the initial state in the second Grandjean zone, as well as in higher-order zones, is associated with the formation of local topologically equivalent states. In these states, the helix has a reduced integer number of helix half-turns throughout the layer thickness or unwound into the planar alignment state.

1. Introduction

Cholesteric liquid crystals (CLCs) stand out among other types of LCs because of their ability to spontaneously form a helical director distribution [1]. This feature allows us to consider these liquid crystal structures as one-dimensional photonic crystals with a forbidden photonic stopband [2], which determines their demand for numerous photonic applications, such as tunable optical filters, mirrors, beam deflectors [3,4,5], and, especially, laser elements [6,7]. Lasing was studied by many authors in a typical geometry of a Grandjean texture with a helical axis directed along the normal to the layer plane, under planar alignment conditions at the confining substrates [8,9,10]. Interesting results on lasing were obtained in Cano–Grandjean wedge geometry [11,12]. However, there are limitations related to the longitudinal pumping of CLC-based lasers in planar Grandjean textures [13], which can be avoided by means of the employment of so-called “lying helix” (LH) geometry, with the axis of the cholesteric helix lying in the plane of the layer [13,14,15].

The problem of forming a stable and defect-free LH has been actively discussed since the late 1990s [16]. Over several decades, numerous approaches have been proposed to obtain LH, including the use of periodic anchoring conditions [16], surface relief formed by laser lithography [17], cooling of CLCs from an isotropic phase in an electric field [18,19], or employing photo- or thermally cross-linkable polymers to stabilize LC texture by forming polymer networks [20], microchannels [14], grafts at the confining surfaces [21], and light stimuli [22,23]. It should be noted that the very fact of obtaining LH in most cases was confirmed only indirectly based on the optical texture features observed in a polarizing microscope. In addition, the intrinsic thermodynamic instability of planar helical textures under typical boundary conditions [24] significantly complicates their formation.

We previously showed that a thermodynamically stable LH state can be obtained in an electric field under periodic boundary conditions with binary modulation of the easy axis direction, induced in the alignment layer with an ion beam treatment [15] or photoalignment [25]. To note, in [15,25], one of the planarly aligned cell surfaces was patterned with homeotropic-alignment stripes, making the tilt angle periodically change in binary mode (planar homeotropic) along the in-plane axis. For such a transition, the CLC natural pitch p₀ must be less than the period of binary modulation of the boundary conditions. It is important that, in this case, the helix is strongly deformed by the field. Also, the LH state disappears when the field is turned off. In the context of this work, it is also essential that, in the aforementioned studies, CLC has a positive dielectric anisotropy, and the electric field vector is normal to the CLC layer that is directed along the axis of the initial helix. In the case of planar electric field geometry, when the electric field vector E is perpendicular to the helical axis, such an orientation transition does not occur in CLCs with positive anisotropy. The helix either becomes strongly deformed in pulsed fields, maintaining the original pitch [26,27], or, in static electric fields, is unwound with the formation of a defective texture [28].

In [29], the behavior of CLCs with both positive and negative anisotropy in the planar geometry of a spatially periodic electric field was investigated. Using the fluorescence method, the authors visualized a fundamentally different field-induced change in the initial helical director distribution in the interelectrode area, depending on the sign of the dielectric anisotropy. In particular, for CLCs with negative dielectric anisotropy, the authors reported an orientation transition with the appearance of a periodic texture near one of the surfaces and interpreted this as a rotation of the axis of the cholesteric helix by 90°. This result seems to be very important and was confirmed by the results of the current study.

A few other studies in this area illustrate highly inhomogeneous textures observed in an electric field, which is apparently due to the significant thickness of the layers studied in comparison with the natural pitch of the helix [30,31,32,33,34].

In this work, we studied in detail the behavior of CLCs with negative dielectric anisotropy in a planar electric field geometry for thicknesses comparable to the helix pitch. These studies were conducted for the first four Grandjean zones. There are two parts below. The first section briefly presents optical observations of electric field-induced orientation transitions using a polarizing optical microscope and the features of relaxation of the induced textures after turning off the field in the region of thicknesses corresponding to the second and fourth Grandjean zones. In the second section, the results of optical observations and features of orientation transitions for the four Grandjean zones are discussed in detail, considering the data obtained by numerical simulation.

2. Materials and Methods

An experimental CLC cell is composed of two glass substrates, one of which bears a system of opaque (chromium) interdigitated electrodes (IDEs) with a period L = 15 μm. The width of the electrode strip was w = 5 µm and the distance between the electrodes was l = 10 µm. Glass without electrodes was used as the opposite substrate. The surfaces of both substrates provided a planar orientation of the CLC in the y-direction, as shown in Figure 1. To obtain a planar alignment, standard technology was used to form polyimide layers on the surface of the substrates, followed by rubbing them with a soft cloth parallel to the electrode strips. In the assembled cell, the rubbing directions on the two substrates are opposite. To study the orientation transitions depending on the thickness of the CLC layer, we used a wedge-shaped cell. The thickness of the CLC layer d varied from 0 to 8 μm, owing to the use of a Teflon gasket on only one edge of the cell.

To prepare CLCs, we used as a matrix a nematic mixture that was developed with the following characteristics: main refractive indices n_⊥ = 1.49, n_‖ = 1.58, low-frequency dielectric constant ε_⊥ = 8.2, dielectric anisotropy Δε = ε_‖ − ε_⊥ = −4.15, and transition temperature to the isotropic state T_iso = 79 °C. To impart chirality, an optically active additive 1,4:3,6-Dianhydro-D-sorbitol-2,5-bis(4-hexyloxybenzoate) with a twisting capacity of 49 μm⁻¹ was used. An additive weight concentration of 0.7 wt. % ensured the natural pitch of the cholesteric helix p₀ ≅ 2.3 µm, which was measured using a wedge-shaped cell with a known thickness at the centers of the resulting Grandjean zones.

In the absence of an electric field, the CLC molecules were oriented planarly, and the helix axis was perpendicular to the surface with the electrodes. In a wedge-shaped cell, where the thickness varies along the x-axis, the pitch of the cholesteric helix cannot change monotonically. Instead, the formation of Grandjean zones occurs, in which the number of half-turns of the helix is fixed and changes strictly by one when moving to the neighboring zone. The helical pitch p in each Grandjean zone varies monotonically only within the zone, for example, in the 2nd zone from p₀ − 0.25p₀ to p₀ + 0.25p₀. At the center of each zone, 2p/p₀ = m, where m is an integer corresponding to the number of the Grandjean zone. In the zero zone, the helical pitch p tends to infinity (the helix is untwisted), and a uniform planar orientation is realized.

Numerical simulations were performed using the LCDTDK v.4.0 software created by one of the authors (S.P.P.). Since the calculation for the experimental geometric parameters of the IDE (L = 15 μm) requires significant computing resources, the calculations were performed for the values of the geometric parameters reduced several times. The period of the IDE array was reduced to L = 3 µm with a distance between the electrodes of 2 µm. The natural pitch of the cholesteric helix p₀ and the thickness of the simulated layer also decreased compared to the experimental samples. In the simulation, the minimum layer thickness was 1 µm, and the maximum was 1.5 µm. The number of half-turns per layer thickness varied in accordance with the number of the Grandjean zone. It should be noted that in the simulation, the natural pitch p₀ was chosen such that the helix in a given zone was stressed (for example, in the case of the 2nd zone, 1.2 p₀ = d; Figure 1). In this case, it was possible to obtain the best agreement with the experiment. Since the optical anisotropy of CLC, determined by the difference in the main refractive indices, is quite small in the experiment (n_⊥ = 1.49; n_‖ = 1.58), then with a significant change in the thickness and, accordingly, the optical retardation (n_‖ − n_⊥)d, the direct comparison of optical patterns for simulated and experimental textures remained possible up to the 4th Grandjean zone.

3. Results

Observations of texture changes in an electric field showed sharp differences between the first and second Grandjean zones. In contrast to the first zone, in the second zone, after turning off the electric field, the induced texture remains for a long time. Similarities between the second and fourth zones were also revealed. For this reason, we begin our discussion with the results of observations in the second and fourth zones.

3.1. Induced Transitions in Second and Fourth Grandjean Zones

Figure 2 shows photographs of field-induced CLC textures in the second (a, b) and fourth (c, d) Grandjean zones at electrical voltages of 50 V and 30 V (rectangular alternating voltage at a frequency of 5 KHz was used), as well as at different time intervals (b, d) after turning off the electrical voltage. The voltage of about 30 V is close to the transition threshold, so the field-induced texture is not well established. At U = 50 V, we deal with the well-established optical texture. It is noteworthy that before the application of electrical voltage, the texture in each zone appeared to be homogeneous in the crossed polarizers, which is typical for Grandjean textures. In an electric field in the interelectrode space, alternating bands of different intensities are observed, oriented along the electrodes. The number of induced bands depends on the number of the Grandjean zone. For example, in the zone with m = 2, two bright stripes are observed, separated between themselves and the electrodes by stripes of lower intensity of transmitted light, as shown in Figure 2a. After turning off the electrical voltage, the induced periodic texture persists for a long time in the form of two bright stripes on a deep black background, as shown in Figure 2b.

In the zone with m = 4, the resulting texture is more defective, as seen in Figure 2c,d. In the electric field, both one bright band and several bright bands were observed. However, after the electric field was turned off, seven bright stripes could be observed in the interelectrode space. Among these bands, the two brightest stand out, located symmetrically relative to the middle of the interelectrode space.

As can be seen from the photo in Figure 2b,d, the resulting periodic textures relax very slowly. Even for the m = 2 zone, where the layer thickness is about 2.5 µm, the periodic texture in the interelectrode space is maintained for several seconds after the electric field is turned off. In the zone with m = 4, the relaxation time increases by tens of times. Thus, after turning off the field, the relaxation time τ cannot be estimated from the well-known relation:

$$\tau ={\displaystyle \frac{\gamma d^2}{{\pi }^{2}K}},$$

(1)

which for the thickness d = 2.5 µm, rotational viscosity γ = 0.1 Pa·s, and effective elastic coefficient K = 10 pN gives the value τ = 6 ms. The periodic texture, for example, in the zone with m = 2 persists for 60 s, and in the zone with m = 4 does not disappear even after 10 min, as seen in Figure 2d. Thus, in the second and fourth Grandjean zones, the appearance of a planar periodic texture can be considered as a field-induced transition to a long-living metastable state.

In the first Grandjean zone, a very rapid relaxation of the induced texture was observed with a characteristic time that approximately coincides with estimate (1). The reason for such rapid relaxation in the first zone is explained below, taking into account the numerical simulation results.

3.2. Simulation of Orientation Transitions and Discussion

The numerical modeling is grounded on the equations of the continuum theory of liquid crystals for finding the spatial–temporal distributions of the LC director, as well as Maxwell’s equations for calculating the distributions of the low-frequency electric and optical fields. The solution to the problem is implemented in the LCD TDK software package (developed by S.P.P.).

In particular, the three-dimensional dynamics of the liquid crystal director n = (n_x, n_y, n_z) is described by the following equations:

$$\begin{gathered} -{\displaystyle \frac{\mathit{\partial }\left(F+g\right)}{\mathit{\partial }{n}_i}}+{\displaystyle \frac{d}{dx}}\left({\displaystyle \frac{\mathit{\partial }\left(F+g\right)}{\frac{\mathit{\partial }{n}_i}{\mathit{\partial }x}}}\right)+{\displaystyle \frac{d}{dx}}\left({\displaystyle \frac{\mathit{\partial }\left(F+g\right)}{\frac{\mathit{\partial }{n}_i}{\mathit{\partial }y}}}\right)+{\displaystyle \frac{d}{dx}}\left({\displaystyle \frac{\mathit{\partial }\left(F+g\right)}{\frac{\mathit{\partial }{n}_i}{\mathit{\partial }z}}}\right)=\gamma {\displaystyle \frac{d{n}_i}{dt}}, \\ i\in \left\{x,y,z\right\}, g\equiv {\displaystyle \frac{1}{2}}\mu \left(1-{{\displaystyle \sum }}_i{n}_i^2\right)=0, \end{gathered}$$

(2)

$$n_x^2+n_y^2+n_z^2=1,$$

(3)

wherein F stands for free energy density that is described as follows:

$$\begin{gathered} F={\displaystyle \frac{1}{2}}\left\{K_1{\left(\nabla \cdot n\right)}^2+K_2{\left(n\cdot \left(\nabla \times n\right)-q_0\right)}^2+K_3{\left(n\times \left(\nabla \times n\right)\right)}^2\right\}-E{P}_{\mathit{f}}-{\displaystyle \frac{{\mathit{\epsilon }}_{\mathbf{0}}}{\mathbf{2}}}\left(\epsilon ·E\right)E, \\ {P}_{\mathit{f}}=e_1\left(\mathbf{\nabla }·n\right)n-e_3\left(n\times \mathbf{\nabla }\times n\right), \end{gathered}$$

(4)

wherein γ is the rotational viscosity, K_1,2,3 are the LC elasticity coefficients, ε is the low-frequency permittivity tensor, the free space dielectric constant ε₀ ≅ 8.85 × 10⁻¹² F/m, E is the electric field vector, and P_f is the flexoelectric polarization vector determined by the flexoelectric coefficients e_1,3 and the corresponding deformation of the director distribution. In the current work, the simulation results are for P_f = 0. Chirality is determined by the wavenumber q₀, which specifies the natural pitch of the helix p₀ = 2π/q₀. The parameter g and the corresponding Lagrange multiplier µ are related to the unit length of the director n, and are taken into account automatically in the numerical solution, when normalization (3) is performed at each discrete moment of time. Also, in present simulations, we use planar alignment along the y-axis (see Figure 1) at rigid boundary conditions (infinite anchoring).

• Induced orientation transition in the first Grandjean zone

Figure 3a shows the simulation results and experimental photographs of the observed textures in the case of an electric field-induced orientational transition in the first Grandjean zone. Initially, in the absence of electrical voltage, the axis of the cholesteric helix is oriented perpendicular to the xy-plane of the layer and directed along the z-axis. In the first zone (m = 1), only one half-turn of the helix is across the thickness of the simulated LC layer (d = 1 μm). At an electric voltage (U = 26 V) in the interelectrode space, a transformation of the Grandjean texture, uniform in the xy-plane, into a planar-modulated structure occurs. As can be seen from the spatial distribution of the LC director, a helix director distribution with a pitch of 0.95 μm along the x-axis appears in the center of the layer, as shown in Figure 3a. In the interelectrode space between the lines AD and CF, two half-turns of the helix are laid. That is, the observed transition can be characterized as a transition to a state with LH. The transition is local, since above the electrodes, excluding areas at the edges, the original Grandjean texture is preserved. An important feature of the transition is the emergence of special localized areas (below, for simplicity, we will call them special points), clearly visible in the gaps marked in the figure as AD, BE, and CF. If, for example, we move along the line AD, then as it passes through a singular point near the lower surface, the y-component of the director changes sign (in Figure 3a, this is reflected by the color change of the end of the cylinder showing the director from red to blue). That is, as in the original Grandjean texture, we have a rotation of the director by π at the layer thickness along the AD line. Since, almost along the entire length of the segment AD, except the vicinity of the mentioned singular point, the director of the LC is oriented planarly, we will call such a local orientation “quasi-planar”. A similar quasi-planar orientation occurs along the BE and CF lines. However, if the singular point on line BE is located near the upper surface of the layer, then on lines AD and CF, it is near the bottom. Singular points marked in a 2D drawing in 3D space form lines along the y-axis.

In crossed polarizers of a polarizing microscope, when the y-axis is oriented at 45° to the polarizer axes, these three quasi-planar areas along the y-direction appear in the calculated optical image as three bright stripes, as shown in Figure 3b. A completely similar optical picture is observed in the experiment, as shown in Figure 3c. In addition, in the experiment, one can observe the defects noted in Figure 3c with red circles. A discussion of the mechanism of formation of these defects is beyond the scope of this work.

An important feature of the transition in the first Grandjean zone is that when a lying helix is formed, the Grandjean texture with one half-turn in the z-direction is essentially preserved, but at the same time, it is strongly deformed in the z-direction and is inhomogeneous in the xy-plane. As will be shown below, this feature associated with the preservation of a half-turn of the helix throughout the thickness of the layer fundamentally distinguishes the transition in the first zone from transitions in zones with a higher m-number. This also leads to a significantly faster relaxation rate of the induced LH after the field is turned off compared to the relaxation rate in higher-order Grandjean zones, which is also discussed below.

b.

Induced orientation transition in the second Grandjean zone

According to numerical simulation data, the transition in the second Grandjean zone, where two half-turns of the helix are across the thickness of the layer, is fundamentally different from the transition considered above in the first zone. Figure 4a shows the results of modeling the transition at an electrical voltage of U = 40 V and after turning off the electric field. The difference between the induced texture (Figure 4a) and the texture in the first zone (Figure 3) is the appearance of two local regions AD and CF with a truly planar orientation throughout the entire thickness. Thus, in these regions, the initial 2π-state transforms into an untwisted homogeneous planar state. Such a transition is well known as topologically equivalent, i.e., realized as a result of continuous deformation of the director distribution in the volume of the layer. This transition was observed in bistability effects in planarly aligned CLC cells [35,36]. The twisted 2π-state of the second Grandjean zone is topologically equivalent to the homogeneous planar state of the zeroth Grandjean zone. A continuous (defect-free) transition between these two states is possible through the homeotropic (vertical) state of the director at the center of the layer, which, in this case, is induced by the electric field. To achieve this homeotropic state, a certain electric field is required, which is why the transition is characterized by some threshold voltage. Since topologically equivalent states differ in twist angles that are multiples of 2π, topologically equivalent transitions are possible between states characteristic of Grandjean zones with either exclusively even or odd m-numbers. A defect-free transition between states of even and odd zones turns out to be forbidden under planar boundary conditions [37]. Only in the case of hybrid boundary conditions is a field-induced transition possible with a change in the twist angle by π, i.e., between neighboring Grandjean zones [38]. This information is important for understanding the transitions in the third and fourth Grandjean zones.

In an electric field, the AD and CF areas with planar orientation are created along the y-direction and appear in the simulated and experimentally observed optical picture in the form of two wide bright stripes, as seen in Figure 4b. In the center between the electrodes (line BE), the transition to the planar state does not occur. It can be seen that when moving along BE, the y-component of the director changes sign twice, respectively, near the upper and lower surface. That is, in this case, there is a highly deformed quasi-planar state in the electric field. The total change in the angle across the layer thickness along BE is equal to 2π and corresponds to the initial 2π-state characteristic of the second Grandjean zone. In the numerically simulated optical picture (see inset in Figure 4b), this state appears as a narrow bright stripe in the center between the electrodes. This bright and narrow band is also visible in the experimental image. Since in this narrow spatial region, in fact, the original 2π-state, although deformed, is preserved, then after turning off the electric field, this state relaxes very quickly, and the narrow stripe in the image in the center between the electrodes disappears, as seen in Figure 4c. However, the local planar states AB and EF, separated from the 2π-state by an energy barrier [35], are preserved, and two bright stripes between the electrodes remain in the model and experimental images, as seen in Figure 4c. Since, in the second Grandjean zone, local planar states are separated by an energy barrier from the lowest-energy ground 2π-state, their relaxation occurs extremely slowly.

Below, we will also call the induced planar state the 0-state, thereby indicating that it can arise in Grandjean zones of a higher order with a total director twist angle that is a multiple of 2π.

c.

Induced orientation transition in the third and fourth Grandjean zones

It was shown above that a distinct feature of the transition in the second Grandjean zone, compared to the transition in the first zone, is the emergence of topologically equivalent 0-states. In the third and fourth zones, the situation is similar. Since topologically equivalent states are characterized by a change in the twist angle across the layer’s thickness by values that are multiples of 2π, then for the third zone, local states with a twist angle of 3π ± 2πm are topologically equivalent, where m is an integer. That is, the initial director distribution with three half-turns at the layer thickness, characteristic of the third Grandjean zone, corresponds to a topologically equivalent π-state, which is the ground one in the first Grandjean zone. For the fourth zone, two topologically equivalent transitions are possible: a transition to the 2π-state and a transition to the 0-state. It is also important to note that since, in the initial Grandjean texture, the electric field is directed predominantly perpendicular to the helix axis, transitions to states with a less twisted helix are energetically favorable [1], i.e., topologically equivalent transitions occur in the direction with a decrease in the initial number of half-turns over the layer thickness.

What has been said regarding the third zone can be seen in Figure 5a, which shows the simulation results at an electrical voltage U = 50 V. Along the four lines, AE, BF, CG, and DH, transitions to local highly deformed topologically equivalent π-states occur. Since these states are characterized by a quasi-planar orientation, they appear in the optical image as the four brightest stripes in the y-direction, as shown in Figure 5b. Since new topologically equivalent π-states are locally formed during the transition, after turning off the electric field, their relaxation is very slow, and the induced state with a lying helix can be observed for quite a long time, as shown in Figure 5c.

The behavior in the fourth Grandjean zone (m = 4), although it looks more complex, also fully fits into the concept of topologically equivalent transitions.

Figure 6a shows that in an electric field in the center between the electrodes, a topologically equivalent transition from the ground 4π-state to the 2π-state is realized. In optics (both in simulation and experiment), this highly deformed 2π-state appears as a single bright stripe in the center between the electrodes, as shown in Figure 6b. After turning off the electric field, relaxation also proceeds through topologically equivalent states, which appear in optical images in the form of many bands, among which two bands correspond to the 0-state and are therefore the brightest, as shown in Figure 6c. Thus, in the fourth zone, we deal with multiple ways of relaxation to the ground 4π-state from topologically equivalent 2π- and 0-states.

Concluding this section, we generalize the features of orientational transitions to the LH state in four primary Grandjean zones, which can be accompanied by local topologically equivalent transitions. Figure 7 schematically shows the energy diagram of allowed transitions in different Grandjean zones. As can be seen, in the first zone, where the ground state is the π-state, there are no allowed transitions, since the nearest untwisted 0-state is not topologically equivalent. In the second and third zones, only one transition to a state with the change for two half-turns is allowed, and in the fourth zone, there are two such transitions. The scheme shown in Figure 7 can easily be continued for zones with m > 4 taking into account that only transitions between the states with an even number of half-turns are allowed.

d.

Dynamics of induced transitions

Figure 8 shows the simulated dynamics of relaxation of induced states for four Grandjean zones (m = 1, 2, 3, 4), obtained for the same thickness of the LC layer d = 1.5 μm. However, as already mentioned, the natural pitch p₀ for each zone was set to obtain a stressed helix at an LC layer thickness d = mp₀/2 + 0.2p₀, so we shift from a zone center towards the next zone with a higher number m. In this case, we obtain the best agreement with the experimental observations. Calculations were performed for the following CLC parameters: elasticity coefficients K₁ = 13.9 pN, K₂ = 11.4 pN, K₃ = 15.5 pN, and rotational viscosity γ = 0.1 Pa·s. To illustrate the dynamics of transitions, the value of the director z-component n_z in the center of the layer between the electrodes was chosen as a parameter characterizing the induced state.

In the first Grandjean zone (m = 1), the relaxation rate is characterized by the shortest time τ~5 ms. At the same time, for large values of m, the relaxation rate drops sharply, reaching its lowest value in the fourth zone (t > 250 ms). All this is well explained within the framework of the model of topologically equivalent transitions discussed above, as seen in Figure 7. Indeed, for m = 1, transitions to topologically equivalent states do not occur since there is no corresponding topologically equivalent state with fewer half-turns. Therefore, after turning off the field, the usual viscoelastic relaxation takes place with a characteristic time, which can be estimated from relation (1). By substituting the LC parameters into (1), it is easy to obtain a characteristic time of ~5 ms, which is consistent with numerical modeling and experimental observations. For Grandjean zones with m > 1, topologically equivalent states exist, and when a lying helix is induced, corresponding local transitions occur. Thus, for a zone with m = 2, local transitions to the 0-state with a uniform planar director distribution take place. Relaxation of these 0-states into the original 2π-states requires activation of the local homeotropic states in the layer. Such activations are possible through spatial regions adjacent in the x-direction, where across the layer thickness, local director orientation is homeotropic (see the director distribution to the left and right of the AD line in Figure 4a).

In order to interpret a correlation between the relaxation times in the second and third zones, it is important to take into account the difference in the free energy between topologically equivalent states. According to Figure 2, the 0- and π-states for the second and third zones, respectively, have the same free energy, and the difference in the relaxation time, which is shorter for the third zone (85 ms vs 125 ms), looks surprising at first glance. However, the energy levels in Figure 7 are given for an unstressed helix located in the center of Grandjean zones. In general cases, the total free energy of the CLC helix per unit area of a layer (integral of the free energy density over the thickness) at rigid boundary conditions is expressed as follows [37]:

$${\mathrm{\Phi }}_{m,k}={\displaystyle \frac{K_{2}d}{2}}{\left(q_{m,k}-q_{m,0}\right)}^2,$$

(5)

where index m is for the Grandjean zone number; k is for a number of helix half-turns in a given zone m; and q_m,k and q_m,0 are for actual and natural wavenumbers of the helix, respectively. The ground state in the m-th zone is characterized by wavenumber q_m,m, which corresponds to m helix half-turns over the layer thickness. The difference in the free energy between the topologically equivalent states with k half-turns and ground state with m half-turns can be derived from (5) as follows:

$${\Delta \mathrm{\Phi }}_{k,m}={ \mathrm{\Phi }}_{m,k}-{\mathrm{\Phi }}_{m,m}={\displaystyle \frac{K_{2}d}{2}}{\left(\Delta k{q}_{1,1}\right)}^2\left[1+{\displaystyle \frac{2m\left(q_{1,0}-q_{1,1}\right)}{\Delta k{q}_{1,1}}}\right],$$

(6)

where q_m,k = kq_1,1, q_m,m = mq_1,1, and Δk = m-k are taken into account.

According to (6), in the case of the “expanded” helix that we deal with when q_1,0 > q_1,1, the ΔΦ increases with the Grandjean zone number m. This fact explains why, for the third zone (m = 3), the relaxation is faster compared to the second zone (m = 2). However, this seems to work only in simplest cases, when we deal with a single topologically equivalent transition like in the case of the second and third zone, as shown in Figure 7.

In the case of the fourth zone, two topologically equivalent states (0- and 2π-state) can exist, and the relaxation has three possible ways: (i) 0-state → 4π-state; (ii) 0-state → 2π-state→4π-state; and (iii) 2π-state → 4π-state. This is why the relaxation process turns out to be more complicated and slowest among the first four Grandjean zones.

4. Conclusions

In a CLC layer with negative dielectric anisotropy, local transitions induced by a periodic electric field from planar Grandjean textures with the helix axis normal to the layer into local states with a lying helix were studied experimentally and numerically. For transitions in the Grandjean zones with the number of half-turns at a layer thickness of more than one, the induced new states with the helix axis lying in the plane of the LC layer are metastable and characterized by slow relaxation from several seconds to tens of minutes, depending on the Grandjean zone and layer thickness.

Numerical modeling has shown that for Grandjean zones with m > 1, the formation of a lying helix is associated with multiple local transitions to topologically equivalent states, characterized by a change in the director twist angle by values that are multiples of 2π. An important circumstance is that, in contrast to topologically equivalent transitions observed previously [35,36], there is no need to involve hydrodynamic flows.

Electric field-induced lying helix states are interesting not only for their topological features but also as superperiodic structures for electro-optical and photonic applications. For example, their properties, characteristic of photonic crystals, can manifest themselves in the waveguide mode, and the topologically equivalent transition to an unwound planar state is of interest for new types of bistable switching.